, for any finite group.
Actually such strong bounds are not even true in many interesting cases: see Example 5.4.1, which shows that diam(Alt(n)2) = Ω(n2) whereas log|Alt(n)2| ≤ 2nlogn (with the same methods we can prove the same lower bound for Alt(n) itself). Another example would be G= (Z/2Z)n, which cannot be generated by less thatnelements: the setA={0n} ∪ {0i10n−i−1|0≤i < n}however generates Ginnsteps, so that diam(G) = Ω(log|G|). In the latter example, we are hindered by the abelianness ofG, which stifles the growth ofAsince not all products ofk elements are distinct; case in point, we have seen Z/nZhaving linear diameter.
Many finite groups are, however, not abelian. In fact, most of them are not:
the number of abelian groups of order≤nis linear in n(see [Ivi85, Thm. 14.6]), while the number of groups of orderpnis at leastp(272−o(1))n3forpfixed prime and n→ ∞, as was already known to Higman [Hig60]. For non-commutative groups, one can hope and often expectAto coverGin a shorter number of steps, and the Cayley graph Cay(G, A) or evenG itself to have a relatively small diameter. In the next sections, we are going to introduce the concept of simple group and see what has been proved or conjectured about their diameters.
1.2 Finite simple groups
LetGbe a group, not necessarily finite. Among all its subgroups,normal subgroups occupy a special place: these are the subgroups N such that gN =N g for any g∈G, and the property of being normal is denoted by the notationNEG. Normal subgroups enjoy many nice properties, first and foremost the fact that we do not have to be careful about left and right multiplication, which is used to show that the set of (left or right) cosets ofN behaves like a group as well, thequotient group G/N.
One could expect to be able to understand G, at least to a certain extent, by studyingN and G/N instead: a spectacularly appropriate example of this is provided by Lemma 6.2.5. Hence, we could and should aim to reduce ourselves to the smallest possible unit of study.
Definition 1.2.1. LetGbe a group. Gis said to be simple if there are no normal subgroups NEGother thanN ={e} andN =G.
We have claimed somewhat vaguely that simple groups are the smallest objects worth studying in some contexts. Their role is often compared to the one played by prime numbers in the context of integers: we can split a number n into two factorsa, bsuch thatn=ab, and then keep going until we end up with a product
of prime numbers. Of course, we are making use of the fact thatZis a UFD, i.e.
that there is an essentially unique way to splitninto its prime factors; if we want to keep up this giuoco delle parti, we need an analogous result in the context of groups. Fortunately, at least for finite groups, we do have such a result.
Definition 1.2.2. Let Gbe a finite group. A composition series is a finite chain of proper normal subgroups
{e}=H0CH1CH2C. . .CHn−1CHn=G
such that the quotient Hi/Hi−1 is simple for every 1 ≤ i ≤ n. The quotients Hi/Hi−1 are called composition factors.
Theorem 1.2.3 (Jordan-H¨older theorem). Let G be a finite group. Then, any two composition series ofGare equivalent, i.e. they have the same length and the composition factors that appear in the two series are the same up to isomorphism and up to permutation of their position in the series.
This theorem is named after Jordan, who proved that the quotients have the same size up to permutation [Jor70,§55]3, and H¨older, who proved that they are actually isomorphic [H¨ol89]; see [Bau06] for a short, modern proof. It is worth noting that the theorem is generalizable to infinite groups and transfinite series, as long as they are ascending and not descending: see [Bir34, Thm. 1] for a proof of the ascending case and [Bir34, Thm. 2] for a counterexample of the descending one.
Thanks to Jordan-H¨older, the study of finite groups can often reduce to the study of finite simple groups instead. The first question that comes to mind then is: which are the finite simple groups?
The history of the search for a definitive answer to this question is quite ar-ticulate, and in some sense still ongoing. Solomon [Sol01] offers a good overview;
we will give here only a handful of highlights. The concepts of normal subgroup and of simple group go back to Galois, who famously proved that Alt(n) is simple for n ≥5 in order to show that general equations of degree ≥5 are not soluble through radicals [Gal46b]; he later proved that PSL(2, p) is also simple for primes p >3 [Gal46a]. An actual conscious search for all finite simple groups is conven-tionally believed to have started with a question by H¨older [H¨ol92]. The work of classifying all such groups went on for almost a century after that.
The period of most intense advancement is generally considered to have be-gun in 1955, when the Brauer-Fowler theorem [BF55] showed a concrete way of attacking the problem through the study of centralizers of involutions: on this same track lies the Feit-Thompson theorem, whose proof appeared eight years later [FT63]. In the years between 1976 and 1983 the classification project was essentially wrapping up, and in this period it was declared to be near completion or completed by several mathematicians, like Brauer [Bra79], Collins [Col80] and
3In 150 years, the language has changed: a “substitution” is an element of the group [Jor70,
§23], and “permutable” means normal [Jor70,§35]. The groups that Jordan was studying were permutation groups, whence the terminology; the proof itself does not restrict only to these groups, though.
Gorenstein [Gor82]; it was chiefly Gorenstein’s announcement that marked the moment when the project was believed to have reached its completion. However, the case of quasithin groups had been solved only partially, in an unpublished manuscript by Mason that still had gaps to be filled. Aschbacher and Smith fixed this last important missing piece only in 2004 [AS04a] [AS04b].
From that year onwards, the Classification of the Finite Simple Groups (CFSG) has generally been accepted to be proved. Here is the statement.
Theorem 1.2.4(CFSG). For any groupG,Gis a finite simple group if and only if it is one of the following:
(a) a cyclic groupZ/pZ withpprime (the only abelian groups of the list);
(b) an alternating group Alt(n)forn≥5;
(c) a group of Lie type among the following 16 families (in all of them,qis a prime power): PSLn(q)(with n≥2and(n, q)6= (2,2),(2,3)),PSUn(q)(withn≥3 and (n, q) 6= (3,2)), PSp2n(q) (with n ≥ 2 and (n, q) 6= (2,2)), PΩ2n+1(q) (with n≥3 andq odd), PΩ+2n(q)(with n≥4),PΩ−2n(q) (withn≥4),G2(q) (with q 6= 2), F4(q), E6(q), 2E6(q), 3D4(q), E7(q), E8(q), 2B2(22n+1) (with n≥1),2G2(32n+1)(with n≥1),2F4(22n+1)(withn≥1);
(d) one of 26 sporadic groups (M11, M12, M22,M23,M24,Co1, Co2,Co3,McL, HS, Suz, J2, Fi22,Fi23, Fi024, M, B, Th,HN,He, J1, J3,J4,O0N, Ly,Ru) or the Tits group2F4(2)0.
For the notation, see [Wil09,§1.2]; mind that there is a finite number of repe-titions in points (b) and (c) of the list.
The proof of Theorem 1.2.4, as it stands today, is distributed across hundreds of articles that total around 10000 pages: this is chiefly the reason why most people refer to CFSG as “widely accepted” instead of saying straight up “a theorem”, for its unwieldy proof is not fit for human consumption. The truth is, mathematics is still on some extent based on trusting the community of mathematicians: while in principle it is a game of absolute rigour, humans are humans and may make mistakes in writing and proofreading depending on whether they have skipped lunch a certain day4. The author eventually learned to accept this fact5, and length is for sure not a sufficient reason for making a proof not a proof, certainly not when the proofreading machines that we are last at most 122 years and 164 days6: thus, for us CFSG is Theorem 1.2.4, emphasis on “Theorem”.
4[E]t idem / indignor, quandoque bonus dormitat Homerus: / verum operi longo fas est obrepere somnum. (Quintus Horatius Flaccus,Ars Poetica, 358-360)
5The healthiest attitude towards this problem on the part of a scientist, as far as the author has encountered, is expressed by Stephen Jay Gould, the paleontologist: faced with a change of the consensus on a particular scientific issue which was outside his expertise but affected his own conclusions, he had to “acknowledge, and [...] provisionally accept” (Gould,The Structure of Evolutionary Theory,§9.3.2). If “provisionally”, for whatever reason, extends until we reach death or retirement or other invalidating circumstances, a problem that Gould does not address, the author (in a beautiful Italian turn of phrase that is coincidentally appropriate on multiple levels) acceptscon filosofia.
6Jeanne Calment.
In any case, there is an ongoing process of writing a second-generation proof: as of the time of writing, 8 volumes out of the planned 13 have been published [GLS94]
[GLS96] [GLS98] [GLS99] [GLS02] [GLS05] [GLS18a] [GLS18b]. Moreover, there has been a computer verification of an important part of CFSG, namely the Feit-Thompson theorem has been proved using Coq, a theorem-proving software (see [GAA+13]).
Let us leave the topic of CFSG itself, however interesting its history and philo-sophical implications may be, and move to one of its consequences that will be important to us. It is a classification of primitive subgroups of Sym(n); the ver-sion below is due to Mar´oti [Mar02], but the original result comes from Cameron [Cam81]. For the definition ofk-transitive, primitive and wreath product, see§3.1.
Theorem 1.2.5. Let n≥1 and let G≤Sym(n) be primitive. Then, one of the following alternatives holds:
(a) there are integers m, r, k such that Alt(m)r ≤G ≤Sym(m)oSym(r), where Alt(m)acts onk-subsets of{1,2, . . . , m}and the wreath product action is the primitive one (so that in particularn= mkr
);
(b) Gis one of the sporadic groupsM11,M12,M23,M24with their4-transitive ac-tion;
(c) |G| ≤nQblog2nc−1
i=0 (n−2i)< n1+log2n.
Even the history of this particular result is quite involved. The first version of a classification of primitive permutation subgroups like the one above appeared in 1981 and was due to Cameron [Cam81, Thm. 6.1]; the proof depends on CFSG (whose statement was already known and considered likely to be correct at the time, and it is referenced to as a “hypothesis” in [Cam81,§1]) and on the O’Nan-Scott theorem. The latter result appeared first in an article for a 1979 conference by Scott [Sco80], who stated in a footnote that O’Nan had also independently obtained it: the theorem, which does not depend on CFSG, offers a classification of maximal permutation subgroups. However, the O’Nan-Scott theorem itself was incorrectly proved: one case, the “twisted wreath action” case (in the language of [LPS88]), was omitted; this has no consequence on the validity of the statement though, as the groups that arise from this case are not maximal. The proof was first corrected by Aschbacher and Scott [AS85], after Cameron’s article had already appeared (as the authors themselves point out)7. Cameron’s original theorem thus is in the unusual position of having been deduced from two major results whose statements were both correct but whose proofs had both an undiscovered gap at the time.
After Cameron’s version, another appeared due to Liebeck [Lie84]: this version is closer to the kind of result we will need to use, and it already acknowledges both
7Technically the first published correction is in [CPSS83], which was received in 1982 and appeared the following year, but the authors of this article make reference to Aschbacher and Scott’s paper “to appear” (it was received in 1983 and it appeared in 1985). Liebeck [Lie84]
refers to the theorem as being corrected in [CPSS83], but adds “for instance”: this was in 1984, after all.
CFSG as a “theorem” (this was after Gorenstein’s announcement, but before the quasithin gap had been truly acknowledged) and the correction of the O’Nan-Scott theorem. Even later, Mar´oti offered the version stated before (see [Mar02, Thm. 1.1]), which is in some sense the furthest possible refinement of Liebeck’s theorem: if we were to tighten (c) inside Theorem 1.2.5 even further, an infinite family of exceptions as in (b) would emerge.
Now that we have a list of what a finite simple group can be, let us move to the next problem we face, namely what we can state about the diameter of such a group.